Characters of algebras containing a Jones basic construction: The Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras
If \(A\) and \(B\) are split semisimple algebras such that \(A\) is a subalgebra of \(B\), the centralizer of the action of \(A\) on \(B\) by left multiplication, \(C=\text{End}_A(B)\), is called the Jones basic construction for the inclusion \(A\subseteq B\). The paper is devoted to computation of characters of algebras containing the basic construction, namely of Temperley-Lieb, Brauer, Birman-Wenzl and Okada algebras. A general theorem is given which says that the characters of \(C=\text{End}_A(B)\) are completely determined by the characters of the subalgebra \(A\). The characters of irreducible \(C\) are determined in general form and used to compute the characters of the Temperley-Lieb algebras and the Okada algebras. To determine the characters of the Birman-Wenzl (B.-W.) algebras, the theory of quantum groups and the quantum Yang-Baxter equation are exploited. In addition to the characters which are obtained due to the general theorem the others are computed by using the duality between B.-W. algebras and Drinfeld-Jimbo (1986) quantum groups of certain types. Certain characters of the quantum groups are obtained which are polynomials invariant under the Weyl group and can be viewed as generalizations of the power symmetric functions and of certain Hall-Littlewood polynomials. It is shown that the characters of the B.-W. algebras may be derived as the coefficients in the expansions of these polynomials in terms of the Weyl characters. As a result, combinatorial rules for computing the irreducible characters of B.-W. algebras are given which may be also used as special cases for computation of the irreducible characters for Iwahori-Hecke (1964) and the Brauer (Ram, 1991) algebras.
scientific article; zbMATH DE number 863620